Uncertainty analysis of ANN based spectral analysis
using Monte Carlo method
J.R. Salinas1, F. García-Lagos1, J. Díaz de Aguilar2, G. Joya1 and F. Sandoval1
1
Grupo ISIS, Dpto. Tecnología Electrónica, ETSI Telecomunicación, Universidad de Málaga
Campus de Teatinos s/n, 29071 Málaga, Spain
[email protected], [email protected]
2 Centro Español de Metrología (CEM)
Alfar 2, 28760 Tres Cantos, Spain
Abstract. Uncertainty analysis of an Artificial Neural Network (ANN) based
method for spectral analysis of asynchronously sampled signals is performed.
Main uncertainty components contributions, jitter and quantization noise, are
considered in order to obtain the signal amplitude and phase uncertainties using
Monte Carlo method. The analysis performed identifies also uncertainties main
contributions depending on parameters configurations. The analysis is performed
simultaneously with the proposed method and two others: Discrete Fourier
Transform (DFT) and Multiharmonic Sine Fitting Method (MSFM), in order to
compare them in terms of uncertainty. Results show the proposed method has the
same uncertainty as DFT for amplitude values and around double uncertainty in
phase values.
Keywords: sine-fitting methods, spectral analysis, ADALINE, ANN, digital
measurement, uncertainty, Monte-Carlo, DFT.
1
Introduction
The use of nonlinear electronic components connected to the electricity grid it is
becoming more common in our daily life. These components add harmonic and interharmonic content to the electric signal, which results in a deterioration of the quality of
the power supplied, an increase in losses and a decrease in the reliability of the whole
system. In this situation, it is increasingly important to accurately determine the harmonic content of the power signals. For these reasons, several National Metrological
Institutes (NMIs) have implemented methods to measure power in non-sinusoidal conditions [1] [2]. These new methods make use of the versatility of the digital techniques,
especially considering the possibility of obtaining the spectral analysis of the signals of
interest, and are based on the use of traditional algorithms, as DFT. These methods
require synchronous sampling in order to be accurate. To overcome this problem, some
authors [3] [4] have proposed the use of asynchronous sampling combined with the use
of non-synchronous spectral analysis. In particular sine fitting methods [5] have been
extended successfully to the multi-harmonic case [6].
adfa, p. 1, 2011.
© Springer-Verlag Berlin Heidelberg 2011
Alternatively, a new method based on ANN was presented [7]. The method is based
in the implementation of the multi-harmonic sine fitting algorithm [5] by mean of a
multilayer perceptron neural network (MLP). The ANN method, as the sine fitting, has
the advantage with respect to the traditional spectral analysis methods that it does not
require synchronism between generation and sampling, which reduce the complexity
of the hardware implementation. In comparison with the conventional multi-harmonic
sine fitting method, the ANN method, has a simpler implementation and does not require special adjustments of coefficients for convergence, that depend on the type of
signal being analyzed[6]. The propose method has been implemented in the Spanish
electrical power primary standard, at Centro Español de Metrología [9] [10]. Additionally, this approach has the advantage to allows, with reduced complexity, modifications
of the signal model to be analyzed. In this sense, this work was extended recently to
obtain inter-harmonic components in [8].
At a primary level, the traceability of such sampling systems is a complex task due
to the complexity of the tests and validations of involved algorithms. The main contribution of this work is the validation for sinusoidal signals of the ANN based spectral
analysis method proposed in [7] by means of Monte Carlo method. In order to obtain
useful results for real applications, parameters configurations used in laboratory will be
used.
Following sections are structured as follows: in section 2 the system to be characterized is briefly presented; in section 3 the Monte Carlo method applied to the characterization of high precision measurement systems is introduced; methodology of the tests
carried out is described in section 4 and results are reported in section 5; finally, section
6 exposes the conclusions of this work.
2
Network description
Let us consider a signal of interest, y(t), stationary in the range of analysis, formed
by K harmonic frequencies of the fundamental frequency, fac. The signal can be described mathematically in the terms of eq. (1), where y(t) is formed by the sum of the
contributions of the DC component, the fundamental frequency, fac, and its multiples,
fk=k•fac, k [2, K].
𝑦(𝑡) = 𝐶 + ∑𝐾
𝑘=1[𝐴𝑘 · cos(2𝜋𝑘𝑓𝑎𝑐 𝑡) + 𝐵𝑘 · sin(2𝜋𝑘𝑓𝑎𝑐 𝑡)]
(1)
The proposed ANN based method [7], MANNFM (Multiharmonic ANN Fitting
Method), performs the spectral analysis of a steady state periodical signal composed by
harmonics as described in equation (1). This is done by means of the multilayer neural
network of Figure 1.
Input data to the ANN are the N time instants - t[n], n 𝜖 [0, N-1]- at which the samples of y(t), y[n], are taken. The time vector, t, is scaled by the factor 2·π in order to
generate the arguments of cosines and sines of eq. (1).
The network implements the K order Fourier series synthesis equation. Briefly,
Layer 1, implemented by means of an ADALINE neuron without bias and linear transfer function, has the object to implement the scale factor of the fundamental frequency,
fac, in the arguments of the cosines and sines of the synthesis equation. The output of
Layer 1 is connected in parallel to Layers 2 and 3, both of which are implemented by
K ADALINE neurons with cosine and sine transfer functions respectively. The position
of each neuron in the array of neurons determines its weight value, so that the k scale
factor of each harmonic in the cosine and sine arguments is implemented by this weight.
As a result, the input to the transfer functions of the neurons of Layers 2 and 3 are the
completed arguments of the cosines and sines of the Fourier synthesis equation.
Layer 2
...
Layer 1
A’1
Neurons with linear
transfer function
C
K
2π·t[i]
L
C
1
C
A’K
L
y'[i]
L
f1
1
...
Layer 3
K
Neurons with cos(x)
transfer function
S
B’1
S
S
C’0
B’K
1
Layer 4
Neurons with sin(x)
transfer function
Non fixed weight
Fixed weight
Fig. 1. The ANN architecture proposed [7].
Consequently, the input from the harmonic section to Layer 4 are the arrays
cos(2··k·fac·t), k [1, K] and sin(2··k·fac·t) , k [1, K]. Layer 4 implements, finally,
the sum of the synthesis equation, by means of 2·K neurons with linear transfer functions. The weights of the neurons connected to Layer 2 are identified as the Ak coefficients of eq. (1), and the weights of the neurons connected to Layer 3 are identified as
the Bk coefficients of eq. (1). In addition, Layer 4 includes the DC component of the
Fourier synthesis series, C, by means of its bias value, which is available in this layer.
Therefore, at this point, the output of Layer 4 is equivalent to the output of the eq. (1),
y(t), in the time instants t[n], n [0, N-1].
As mentioned above, this high precision algorithm has been implemented and tested
in the measurements system of the Spanish NMI as part of the electrical power primary
standard.
3
Propagation of distributions and simulation using the Monte
Carlo method
As mentioned previously, due to the complexity of the MANNFM uncertainty evaluation, the MCM is used. In this section the basis of the MCM and its application to
this purpose are presented. The Guide to the Expression of Uncertainty in Measurement
(GUM) [11] provides the framework for uncertainty evaluation, which has three main
stages: formulation, propagation and summarizing. At the Formulation stage the N input quantities, X = (X1, …, XN) upon which the output quantity, Y, depends, are determined and their probability distribution functions (PDF), are assigned. In the Propagation stage, the PDFs for the Xi are propagated through the measurement model, Y= f(X),
in order to obtain the PDF of the output Y. Finally, in the Summarizing stage, using de
PDF obtained for Y, the expectation of Y, y, and its standard deviation, u(y), are obtained1.
The propagation of distributions can be implemented analytically if a mathematical
representation of the PDF for Y can be obtained. However, in practice this option is
possible in simple cases and for more complex situations approximations based on Taylor series must be applied. When these approximations are not possible either (e.g.,
complex or nor linear models) numerical methods must be used.
The base of MCM [12], Figure 2, is to sampling the PDF for Y, obtaining M values
yr, for r =1, ..., M. Each yr is obtained by sampling at random from each of the PDFs
for Xi, and evaluating the model at the samples values so obtained. In case of several
output variables [13], the foundation is the same but Y is a vector provided by a vector
function or model f(X).
x1, u(x1)
x2, u(x2)
Y = f(X)
y, u(y)
x3, u(x3)
Fig. 2. MCM propagation of distributions for 3 independent input quantities
Consequently, in order to apply MCM, it is only necessary that Y can be formed from
the values of X. Although Monte Carlo method is used as a mean to provide a numerical
representation of the PDF for Y, it is also a simulation process that provides information
about the input/output PDFs relationship of a system modeled as f(X). So, the method
can be used not only to obtain output uncertainties of a measurement, but also to simulate a system [14] and evaluate the impact of different parameters as well as the impact
of the different uncertainty contributions in the output uncertainties. The second approach will be used in this work.
4
Methodology applied
4.1
Simultaneous comparison of three methods
In order to compare MANNFM results with those obtained by other methods, the
same tests have been applied to other two methods: DFT [15] and MSFM [6]. DFT can
be considered as a reference, given that, in ideal conditions, provides the spectrum of
the signal under analysis without error. When synchronous sampling is not possible,
1
Although there is a coincidence between some terms used in GUM and this work nomenclatures, since there is no possibility of confusion, GUM nomenclature has been maintained in
this section in order to simplify its reading.
fitting methods are used [3] [5] in order to avoid DFT errors due to spectral leakage.
As MANNFM, MSFM is a fitting method, but solved in a conventional way, solving a
non-linear equation system iteratively.
As can be observed in Figure 3, for comparison purposes the same signal, x(t), will
be sampled and applied to the three methods. The DFT will process synchronous samples of x(t), xs[n], while MSFM and MANNFM will process asynchronous samples of
x(t), xa[n], obtained in a set of time instants ta[n], where ta[n]=t[n]·(1+ ξa), being ξa the
error of synchronism (relative) between generation and sampling.
x(t)
SAMPLING
SAMPLING
t[n]
ASYNC.
xs[n]
xa[n]
DFT
XD(f)
MSFM
XS(f)
MANNFM XN(f)
ta[n]
Fig. 3. Scheme for the comparison of the three methods
4.2
Uncertainties contributions considered
Two uncertainty components will be considered in this work, jitter contributing to
time uncertainty of each sampled point, the sampling jitter; and quantization contributing to a voltage uncertainty of each sampled point. . Both of them have been theoretically studied for DFT [16] [17].
Regarding jitter, a rectangular distribution has been considered. Taking into account
the specifications of usual laboratory instruments, two jitter values (maximum) will be
contemplated: 3·10-12 s and 1·10-7 s. The first value is the jitter specification for the
National Instruments 5922 digitizer, and the second one is the jitter specifications for
the Keysight 3458A digital multimeter used in [9] [10].
Concerning the quantization, it has been simulated taken into account three usual
laboratory resolutions or number of bits, Nbits, values: 16, 18 and 21 bits, which correspond again with the HP 3458A resolution derived from the aperture time ranges configured for high, medium and low sampling frequency respectively. Regarding the full
scale parameter, 1.2 V has been used, which is the value for the 1 V range of the HP
3458A in digital operation mode.
In order to evaluate independent and jointly the impact of both contributions to the
total uncertainty of the three methods, the scheme presented in Figure 3 will be simulated with three different sampling situations: (i) samples affected only by quantification; (ii) samples affected only by jitter; and (iii) samples affected by quantification and
jitter. Terms q, j and jq respectively will be used in reference to signals and results
related to this three sampling situations.
4.3
Parameters values considered
Several parameters values (according to real values used in laboratory) of the signal
to be sampled, have been considered in order to evaluate their impact to the estimated
uncertainty. A sinusoidal signal has been consider, with fundamental frequency, fac, 53
Hz and amplitude 1 V. Concerning to the phase of the sampled signal, uniformly distributed random values between ±π radians have been generated.
Regarding the ratio R=fs/fac, being fs the sampling frequency, three values have been
selected, around 24, 124 and 261, related to low, medium and high sampling frequencies, respectively.
Concerning the number of samples, N, it has to be considered that (i) the Keysight
3458 internal memory maximum capability (for maximum precision) is 37888 samples
and (ii) N and M should not have common factors in order to obtain maximum information of the sampled signal. Consequently, three N values (1311, 7073 and 15457),
related to low, medium and high number of samples have been selected and used in
each ratio value. Being M the number of periods involved in the measurement, R can
be also expressed as R=N/M. So, variations of N keeping R constant implies also to
vary M. Table 1 shows the values of R, N and M considered. In order to assure the DFT
conditions, for N and M integer values have been selected (which imply light variations
of the ratio R obtained).
N
H
M
L
H
M
L
H
M
L
15457
7073
1311
15457
7073
1311
15457
7073
1311
M
R
59
27
5
125
57
11
625
286
53
261.983
261.963
262.200
123.656
124.088
119.182
24.731
24.731
24.736
fs
H
M
L
Table 1. Sampling frequency parameters values considered to estimate the uncertainty of the
system.
Finally, about the error of synchronism, ξa, random values up to ±20%, uniformly
distributed, have been considered. The Interpolated FFT (IpFFT) [18] algorithm has
been used in order to obtain initial estimation of the fundamental frequency for the
asynchronous methods. No convergence problems have been detected in any simulation.
4.4
Computational hardware used
In order to perform the MCM in the different configurations established, large computational resources are required. For this work, resources provided by the University
of Málaga‘s supercomputing center (UMSC) have been used. The computational resources are based in the Cluster Intel E5-2670, with shared memory machines with 2
TB of RAM each and AMD Opteron 6176 and ESX virtualization clusters.
5
Results
All tests have been performed in the same way: with the selected parameters and the
uncertainty contributions defined in section 4.2, 10 6 trials of the MCM has been performed over the three methods (as described in section 4.1) in order to obtain a numerical representation of their PDF outputs.
All the obtained PDFs are Gaussian. The values have been expressed in deviations
to nominal (absolute) and those deviations have been fitted to a normal PDFs by means
of Matlab© normfit function, obtaining their mean and standard deviation, σ. For each
output of each method, mean represents the deviation to nominal and standard deviation
its uncertainty for a confidence level of 62.8%. In next subsections we will present the
most significant results of the tests performed, in terms of mean and standard deviation.
Methods comparison
A first test was performed in order to evaluate similarities between methods outputs
and, at the same time, evaluate independent and jointly the impact of jitter and quantization to total uncertainty. Maximum jitter was fixed to 100 ns and Nbits was set to 16.
The ratio was set to low and 15457 samples were considered (M=625). Tables 2, 3 and
4 show results considering respectively jitter, quantization and both effects on the three
methods, for amplitude and phase of the fundamental frequency.
DFT
MPSF
MANNFM
Ej{ΔA1}
(µV)
1.06E-04
-1.09E-04
-1.11E-04
σj (ΔA1)
(µV)
5.50E-02
5.49E-02
5.49E-02
Ej{ΔΦ1}
(µrad)
6.69E-04
-6.22E-04
-6.21E-04
σj (ΔΦ1)
(µrad)
9.55E-02
1.89E-01
1.89E-01
Table 2. Mean (E) and standard deviation (σ) of fundamental frequency deviations to nominal
amplitude (ΔA1) and phase (ΔΦ1) for three methods - only jitter considered
DFT
MPSF
MANNFM
Eq{ΔA1}
(µV)
6.25E-04
1.16E-03
1.15E-03
σq (ΔA1)
(µV)
1.12E-01
1.12E-01
1.12E-01
Eq{ΔΦ1}
(µrad)
1.26E-04
-1.28E-03
-1.28E-03
σq (ΔΦ1)
(µrad)
1.12E-01
2.41E-01
2.41E-01
Table 3. Mean (E) and standard deviation (σ) of fundamental frequency deviations to nominal
amplitude (ΔA1) and phase (ΔΦ1) for three methods - only quantization considered
DFT
MPSF
MANNFM
Ejq{ΔA1}
(µV)
7.31E-04
1.05E-03
1.05E-03
σjq (ΔA1)
(µV)
1.25E-01
1.33E-01
1.33E-01
Ejq{ΔΦ1}
(µrad)
7.94E-04
-1.91E-03
-1.91E-03
σjq (ΔΦ1)
(µrad)
1.53E-01
3.06E-01
3.06E-01
Table 4. . Mean (E) and standard deviation (σ) of fundamental frequency deviations to nominal
amplitude (ΔA1) and phase (ΔΦ1) for three methods - jitter and quantization considered
Several conclusions can be reached with these results. Firstly, the mean deviations
to nominal for amplitude and phase are around zero, always lower to 2·10 -9 volts or
2·10-9 radians. These values have been repeated with the rest of R, N, jitter and Nbits
values. So, from now, we will present only standard deviation values will be presented.
Additionally, for the three methods, the phase standard deviation is lightly higher than
amplitude standard deviation.
Concerning the fitting methods, results show that both provide almost identical results, which is logical, due both methods solve the same optimization problem in different ways. And comparing DFT and fitting methods, both provides very similar results in amplitude, with practically identical standard deviation, but higher standard
deviations (in a factor around 2) is obtained in phase by fitting methods.
Relating to jitter and quantization impact in the three methods results, for the values
selected, although both affect to the total standard deviation, quantization has more impact than jitter. On the other hand, results show that their jointly impact is equivalent
to the quadratic sum of the independent impacts. That can be explained because although quantization is an uncertainty over the amplitude value and jitter is an uncertainty over the time instant of sampling, jitter is in the last part a noise over the amplitude [1], so both contributions apply, at the end, to the same variable.
Impact of R and N in results
In order to evaluate R and N impact on the methods results, for each combination of
jitter and number of bit values, the 9 points from Table 1 were simulated. Tables 5 and
6 show standard deviation results obtained for amplitude and phase of the fundamental
frequency in the case of jitter value of 100 ns and 16 bits of quantization. In order to
unify data as much as possible, both tables simultaneously present values obtained considering only jitter (j), only quantization (q) and both (jq).
DFT
MSFM
MANNFM
24
1311
R
123
262
24
N
7073
R
123
262
24
15457
R
123
q
4.18E-01
4.18E-01
4.18E-01
1.86E-01
1.86E-01
1.86E-01
1.12E-01
1.12E-01
1.12E-01
j
1.88E-01
1.87E-01
1.88E-01
8.08E-02
8.08E-02
8.09E-02
5.47E-02
5.47E-02
5.47E-02
jq
4.60E-01
4.60E-01
4.59E-01
1.82E-01
1.82E-01
1.82E-01
1.25E-01
1.25E-01
1.24E-01
q
4.14E-01
4.09E-01
4.15E-01
1.81E-01
1.77E-01
1.80E-01
1.20E-01
1.21E-01
1.21E-01
j
1.88E-01
1.88E-01
1.89E-01
8.08E-02
8.08E-02
8.08E-02
5.47E-02
5.46E-02
5.46E-02
jq
4.55E-01
4.53E-01
4.56E-01
1.97E-01
1.95E-01
1.96E-01
1.33E-01
1.32E-01
1.32E-01
q
4.14E-01
4.09E-01
4.15E-01
1.81E-01
1.77E-01
1.80E-01
1.20E-01
1.21E-01
1.21E-01
j
1.88E-01
1.88E-01
1.89E-01
8.08E-02
8.08E-02
8.08E-02
5.47E-02
5.46E-02
5.46E-02
jq
4.55E-01
4.53E-01
4.56E-01
1.97E-01
1.95E-01
1.96E-01
1.33E-01
1.32E-01
1.32E-01
262
Table 5. Standard deviation of fundamental frequency amplitude deviations to nominal values
(µV) for three methods in Table 1 points, considering: only jitter (j), only quantization (q) and
both (jq)
DFT
MSFM
MANNFM
24
1311
R
123
262
24
N
7073
R
123
262
24
15457
R
123
q
4.14E-01
4.14E-01
4.14E-01
2.12E-01
2.12E-01
2.12E-01
1.21E-01
1.21E-01
1.21E-01
j
3.25E-01
3.25E-01
3.25E-01
1.40E-01
1.40E-01
1.40E-01
9.47E-02
9.47E-02
9.47E-02
jq
5.27E-01
5.26E-01
5.26E-01
2.26E-01
2.26E-01
2.27E-01
1.53E-01
1.53E-01
1.53E-01
q
8.27E-01
8.24E-01
8.19E-01
3.54E-01
3.56E-01
3.54E-01
2.41E-01
2.40E-01
2.41E-01
j
6.50E-01
6.51E-01
6.55E-01
2.80E-01
2.80E-01
2.80E-01
1.89E-01
1.89E-01
1.89E-01
jq
1.05E+00
1.05E+00
1.06E+00
4.53E-01
4.53E-01
4.53E-01
3.06E-01
3.06E-01
3.06E-01
q
8.27E-01
8.24E-01
8.19E-01
3.54E-01
3.56E-01
3.54E-01
2.41E-01
2.40E-01
2.41E-01
j
6.50E-01
6.51E-01
6.55E-01
2.80E-01
2.80E-01
2.80E-01
1.89E-01
1.89E-01
1.89E-01
jq
1.05E+00
1.05E+00
1.06E+00
4.53E-01
4.53E-01
4.53E-01
3.06E-01
3.06E-01
3.06E-01
262
Table 6. Standard deviation of fundamental frequency phase deviations to nominals values
(µrad) for three methods in Table 1 points, considering: only jitter (j), only quantization (q) and
both (jq)
Firstly, tables 5 and 6 confirm in a more general way conclusions obtained in previous sections about methods comparisons. Moreover, from both tables it can be observed
that for three methods and three cases j, q and jq, the R value does not affect the standard
deviations of amplitude and phase, while the N value impacts clearly in the results, so
that lower N values produces higher standard deviations. The same behaviour can be
observed for the rest of jitter and Nbits values combinations.
Impact of jitter and number of bits on results
In order to evaluate jitter and resolution impact in the three methods, each combination of Maxjit and Nbits values defined in section 4.2 were considered for simulation.
Regarding R, taking into account previous test results, a single value (the intermediate)
was fixed. Finally, the three N (and M) values were additionally considered, in order to
observe the N influence.
So, 18 simulation points were performed. In all of them, results obtained consolidate
previous tests conclusions: the fitting methods provide identical results which, at the
same time respect DFT results, are identical in amplitude and higher by a factor around
around 2 in phase sigma. For this reason and with the aim of synthetize, only
MANNFM results will be presented in this section, but all conclusions obtained can be
applied to DFT and MSFM.
Results for MANNFM are presented in Tables 7 and 8, which show standard deviations obtained for amplitude and phase of the fundamental frequency at any simulation
point. In order to unify data as much as possible, both tables simultaneously present
values obtained considering only jitter (j), only quantization (q) and both (jq).
First of all, again previous conclusions are confirmed in more general test: as in
previous section, higher N values results in lower standard deviations for both only jitter
and only quantization.
Maxjit
3 ps
Maxjit
100 ns
16
2.42E-06
N
7073
Nbits
18
2.42E-06
21
2.42E-06
j
16
5.68E-06
1311
Nbits
18
5.68E-06
q
3.94E-01
1.03E-01
1.27E-02
1.76E-01
4.49E-02
5.54E-03
1.18E-01
2.96E-02
3.84E-03
jq
3.94E-01
1.03E-01
1.27E-02
1.76E-01
4.49E-02
5.55E-03
1.18E-01
2.96E-02
3.84E-03
j
1.89E-01
1.89E-01
1.89E-01
8.05E-02
8.05E-02
8.05E-02
5.48E-02
5.48E-02
5.48E-02
q
3.94E-01
1.03E-01
1.27E-02
1.76E-01
4.49E-02
5.54E-03
1.18E-01
2.96E-02
3.84E-03
Jq
4.51E-01
2.16E-01
1.90E-01
1.95E-01
9.26E-02
8.07E-02
1.31E-01
6.29E-02
5.50E-02
21
5.68E-06
16
1.64E-06
15457
Nbits
18
1.64E-06
21
1.64E-06
Table 7. Standard deviation of fundamental frequency amplitude deviations to nominal values
(µV) for Maxjit, Nbits and N values simulated, considering: only jitter (j), only quantization (q)
and both (jq)
Maxjit
3 ps
Maxjit
100 ns
16
8.33E-06
N
7073
Nbits
18
8.33E-06
21
8.33E-06
j
16
1.97E-05
1311
Nbits
18
1.97E-05
q
8.68E-01
2.04E-01
2.63E-02
3.53E-01
8.82E-02
1.10E-02
2.36E-01
6.05E-02
7.42E-03
jq
8.68E-01
2.03E-01
2.63E-02
3.53E-01
8.83E-02
1.10E-02
2.36E-01
6.04E-02
7.43E-03
j
6.56E-01
6.56E-01
6.56E-01
2.78E-01
2.78E-01
2.78E-01
1.89E-01
1.89E-01
1.89E-01
q
8.68E-01
2.04E-01
2.63E-02
3.53E-01
8.82E-02
1.10E-02
2.36E-01
6.05E-02
7.42E-03
jq
1.06E+00
6.87E-01
6.56E-01
4.58E-01
2.90E-01
2.78E-01
3.04E-01
1.99E-01
1.89E-01
21
1.97E-05
16
5.68E-06
15457
Nbits
18
5.68E-06
21
5.68E-06
Table 8. Standard deviation of fundamental frequency phase deviations to nominal values
(µrad) for Maxjit, Nbits and N values simulated, considering: only jitter (j), only quantization
(q) and both (jq)
Regarding Maxjit effect on j results, for the lower Maxjit value, 3 ps, we can observe
very low standard deviation values, with maxima (for lower N) of 5.7·10-12 V and
19.7·10-12 rad. On the other hand, for 100 ns, we can observe maxima standard deviations values of 0.2·10-6 V and 0.6·10-6 rad.
Regarding the quantization effect on points selected, we can observe in Figure 4 that
N and Nbits variations have comparable effects on the standard deviations values obtained. This can be useful to compensate the effects of one parameter with the other.
For example, as we can observe in Figure 4, the standard deviation obtained for 18 bits
and 1311 samples can be nearly obtained with 16 bits if 15457 samples are considered.
Finally, regarding the combined impact of jitter and quantization, it can be observed
from tables 7 and 8 that in the low jitter point, impact of jitter is no significant in front
of quantization impact, so that the combined (jq) standard deviations are equal to those
of the only quantization considered case (q).
In the case of Maxjitt equal 100 ns more similar contributions are obtained, so that
both effects contribute to the combined impact. Even so, with the lower N and Nbits
values the main contribution is due to quantization although jitter also contributes. So,
if Nbits raise, at lower N, the quantization contributions decreases; till with 21 bits, the
main contribution is due to jitter. When N value is increased to it next value, the same
behavior is observed, changing the main contribution depending on the Nbits value.
Standard Deviation
(µV)
4.E-01
N
3.E-01
2.E-01
1311
1.E-01
7073
15457
0.E+00
15
17
19
21
Bits of Resolution
Fig. 4. Standard deviation of fundamental frequency amplitude deviations to nominal values
(µV) obtained for Nbits and N values - only quantization considered
6
Conclusions and Future Work
Analysis performed has provided practical values of MANNFM standard deviation
(amplitude and phase of the fundamental) for its use in practical measurements in laboratory. In the worst case, obtained results show uncertainties - for 95.5% of confidence level (2σ) - lower than 1.7 µV and 2.12 µrad for amplitude and phase respectively. From the methods comparison performed, it has been found that fitting methods,
MANNFM and MSFM, provides identical results. Respect DFT, fitting methods are
very similar in amplitude but their standard deviation for the phase of the fundamental
is higher than that of DFT by a factor around 2. Besides that, behavior of fitting methods
regarding uncertainty contributions (jitter sampling and quantization noise) and sampling parameters (R and N) follow the same tendencies than DFT.
Further work is also necessary in order to study uncertainties for fundamental and
harmonics components when signals with harmonic content are sampled. Other frequencies have to be studied also, in order to identify jitter effect when higher fundamental frequencies are necessary.
Acknowledgment
This work was partially supported by the Universidad de Malaga - Campus de Excelencia Andalucia-Tech.
7
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